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Research Papers

An Interval-Based Method for Workspace Analysis of Planar Flexure-Jointed Mechanism

[+] Author and Article Information
D. Oetomo1

Department of Mechanical Engineering, University of Melbourne, Melbourne, VIC 3010, Australia; INRIA Sophia Antipolis, BP93, 06902 Sophia Antipolis Cedex, Francedoetomo@unimelb.edu.au INRIA Sophia Antipolis, BP93, 06902 Sophia Antipolis Cedex, Francedoetomo@unimelb.edu.auDepartment of Mechanical Engineering, Monash University, Clayton, VIC 3800, Australiadoetomo@unimelb.edu.au INRIA Sophia Antipolis, BP93, 06902 Sophia Antipolis Cedex, Francedoetomo@unimelb.edu.au

D. Daney

Department of Mechanical Engineering, University of Melbourne, Melbourne, VIC 3010, Australia; INRIA Sophia Antipolis, BP93, 06902 Sophia Antipolis Cedex, Francedavid.daney@sophia.inria.fr INRIA Sophia Antipolis, BP93, 06902 Sophia Antipolis Cedex, Francedavid.daney@sophia.inria.frDepartment of Mechanical Engineering, Monash University, Clayton, VIC 3800, Australiadavid.daney@sophia.inria.fr INRIA Sophia Antipolis, BP93, 06902 Sophia Antipolis Cedex, Francedavid.daney@sophia.inria.fr

B. Shirinzadeh

Department of Mechanical Engineering, University of Melbourne, Melbourne, VIC 3010, Australia; INRIA Sophia Antipolis, BP93, 06902 Sophia Antipolis Cedex, Francebijan.shirinzadeh@eng.monash.edu.au INRIA Sophia Antipolis, BP93, 06902 Sophia Antipolis Cedex, Francebijan.shirinzadeh@eng.monash.edu.auDepartment of Mechanical Engineering, Monash University, Clayton, VIC 3800, Australiabijan.shirinzadeh@eng.monash.edu.au INRIA Sophia Antipolis, BP93, 06902 Sophia Antipolis Cedex, Francebijan.shirinzadeh@eng.monash.edu.au

J.-P. Merlet

Department of Mechanical Engineering, University of Melbourne, Melbourne, VIC 3010, Australia; INRIA Sophia Antipolis, BP93, 06902 Sophia Antipolis Cedex, Francejean-pierre.merlet@sophia.inria.fr INRIA Sophia Antipolis, BP93, 06902 Sophia Antipolis Cedex, Francejean-pierre.merlet@sophia.inria.frDepartment of Mechanical Engineering, Monash University, Clayton, VIC 3800, Australiajean-pierre.merlet@sophia.inria.fr INRIA Sophia Antipolis, BP93, 06902 Sophia Antipolis Cedex, Francejean-pierre.merlet@sophia.inria.fr

1

Corresponding author.

J. Mech. Des 131(1), 011014 (Dec 16, 2008) (11 pages) doi:10.1115/1.3042151 History: Received February 11, 2008; Revised September 23, 2008; Published December 16, 2008

This paper addresses the problem of certifying the performance of a precision flexure-based mechanism design with respect to the given constraints. Due to the stringent requirements associated with flexure-based precision mechanisms, it is necessary to be able to evaluate and certify the performance at the design stage, taking into account the possible sources of errors such as fabrication tolerances and the modeling inaccuracies in flexure joints. An interval-based method is proposed to certify whether various constraints are satisfied for all points within a required workspace. Unlike the finite-element methods that are commonly used today to evaluate a design, where material properties are used for evaluation on a point-to-point sampling basis, the proposed technique offers a wide range of versatility in the design criteria to be evaluated and the results are true for all continuous values within the certified range of the workspace. This paper takes a pedagogical approach in presenting the interval-based methodologies and the implementation on a planar 3revolute-revolute-revolute (RRR) parallel flexure-based manipulator.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Types of common flexure joints: (a) notch type flexure joint and (b) leaf type flexure joint

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Figure 2

Illustration of the effect of the branch-and-bound on an equality constraint: (a) The original (overestimated) bound of the solution, obtained by interval evaluation. (b) The sharp result with filtering. (c) Branch-and-bound process repeatedly bisects the solution box to a predefined threshold box dimension ϵ to provide a better bound to the solution.

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Figure 3

A 3RRR planar parallel mechanism

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Figure 4

Workspace of the 3RRR planar parallel mechanism. The workspace, constrained by joint limits, after consistency filtering: (a) without considering fabrication tolerances and (b) assuming ±50 μm tolerance on link length r and l. These two dimensional plots are generated at constant θ=θm=−10.3 deg. Allowable flexure-joint deflection is ±3 deg.

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Figure 5

The inner solution admitted into the workspace of the 3RRR planar parallel flexure mechanism. This result takes into account the uncertainties in the kinematic modeling and fabrication tolerances, constrained by the bounds of the allowable flexure-joint deflections. The orientation range is θI=θm±17.5 mrad. The workspace is presented in solid (a) and wire frames (b).

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Figure 6

Singularity free workspace as obtained by evaluation of constraint 16

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Figure 7

Singularity free workspace of the 3RRR planar parallel mechanism, taken at θ=40 deg. The result (a) was obtained by direct evaluation of constraint 16 and is the 2D representation of the result in Fig. 6 at θ=40 deg. The result (b) shows the large improvement provided by the matrix regularity test algorithm, as provided by the ALIAS library. The loci of singular workspace are marked in red (solid color) for clarity.

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Figure 8

Two dimensional workspace at constant θ=θm=−10.3 deg that satisfies the required motion requirement, given the joint space motion resolution. Solving algorithms were (a) preconditioned Hansen–Bliek and (b) symbolic preconditioning with Gaussian elimination.

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Figure 9

Overall workspace due to multiple constraints. (a) The workspace allowable by limits of joint deflection. (b) The workspace that satisfies the required motion resolution. (c) The intersection of all given constraints.

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Figure 10

Usable workspace of the planar flexure-based manipulator for orientation range of −10.3 deg±1 deg

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